Noncommutative Hirzebruch surfaces

By definition, Hirzebruch surface is a $\mathbb{P}^1$ bundle over $\mathbb{P}^1$. Isomorphism classes of such surfaces are classified by non-negative integers d, and those with the same parity are connected by unobstructed deformations.
Flat deformations of the abelian category of coherent sheaves on Hirzebruch surface, or noncommutative deformations, have been studied by several people. Contrary to the case of del Pezzo surfaces, nc deformations of the $d$-th Hirzebruch surface are obstructed if $d > 3$. On the other hand, Michel Van den Bergh introduced the notion of sheaf $\mathbb{Z}$-algebras and proved that any noncommutative deformation of a Hirzebruch surface over a complete Noetherian local ring is obtained from a sheaf $\mathbb{Z}$-algebra associated to a locally sheaf bimodule of rank 2 on the projective line. In this talk, I will give some introduction to this subject and explain our result on the geometric classification of locally free sheaf bimodules. I will also explain some results from the point of view of derived categories, including a version of McKay correspondence. My talk will be based on a joint work in progress with Izuru Mori and Kazushi Ueda.